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## Phys 498A Lecture NotesThursday, January 30, 1997Lecturer: Erik Koch |
HW2 assigned |

If we let k^{2}(x) = (2m/~~h~~) [ E - V(x) ],
the Schrödinger equation takes the form

We want to discretize this to a grid with spacing h. The second derivative operator can be discretized as

where we have used the notation f_{j} = f(x_{j})
with the x_{j}'s being the grid points.

Direct application of the discritized derivative leads to a discretized
Schrödinger equation with errors of order O( h^{4}).

This could be solved for u_{j+1} and used to integrate
the equation.
However, with a little extra work we can get a method that has errors
of order O( h^{6} ), a substantial improvement known as the Numerov
Method.

The discretized 2nd derivative formula is

Thus the Schrödinger equation becomes

Email question/comments/corrections to shumway@uiuc.edu .